Finite Group Databasebeta

← #33108462

Subgroups — #33108462

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988809C211yesmaximal, characteristic
2C2×#17988692C211yesmaximal, characteristic, complemented
3C2×C4×#11924015C211yesmaximal, characteristic
4#22613000C211yesmaximal, characteristic
5#22610745C211yesmaximal, characteristic
6#22610692C211yesmaximal, characteristic, complemented
7#22610627C211yesmaximal, characteristic
8#22610606C211yesmaximal, characteristic, complemented
9#22610286C211yesmaximal, characteristic
10#22610272C211yesmaximal, characteristic, complemented
11#22610198C211yesmaximal, characteristic, complemented
12#22610191C211yesmaximal, characteristic
13#22609806C211yesmaximal, characteristic, complemented
14#22609761C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C73C211yesmaximal, characteristic, complemented
16C22×#11924018C2211yescharacteristic
17C22×#11924015C2211yescharacteristic
18C2×#14556579C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21#17988809C418yescomplemented
22#17988692—24no—
23C4×#11924015—24no—
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×#14556511C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556502C2211yescharacteristic
37C2×C4×#10596831C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988489C2211yescharacteristic
40#17988477—21no—
41#17988476—21no—
42#17988458—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44#17988434—21no—
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988342C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54#389×C73—21no—
55#389×C73—21no—
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#384×C73—21no—
59#384×C73C2211yescharacteristic
60C4×C4⋊C4×C73—21no—
61C4×C4⋊C4×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×D292C2311yescharacteristic
64C22×C4×D73—21no—
65C2×#11924018—22no—
66C2×#11924018—22no—
67C2×#11924018C2×C414yescomplemented
68C2×#11924015—22no—
69C2×#11924015—22no—
70C2×#11924015C2×C414yes—
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75#14556579—22no—
76#14556579C2×C414yes—
77C4×Dic146—22no—
78C4×Dic146—41no—
79C4×D292—22no—
80C4×D292—41no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91#14556522—22no—
92#14556522—22no—
93C4⋊#10596831C2×C414yescomplemented
94#14556515—22no—
95#14556515D414yes—
96#14556514—22no—
97#14556514—41no—
98#14556514C2×C414yescomplemented
99#14556511—22no—
100#14556511—41no—
101#14556511C2×C414yes—
102#14556502—22no—
103#14556502—22no—
104C4×#10596831—22no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×Q8×C73C2311yescharacteristic
109C4×Q8×C73—22no—
110C4×Q8×C73—22no—
111C4×Q8×C73—22no—
112C4×Q8×C73—41no—
113C4×Q8×C73—41no—
114C4×Q8×C73C2×C414yescomplemented
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73D411yescharacteristic
123C2×C4⋊C4×C73D411yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C42×C73—21no—
128C2×C42×C73—21no—
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132C2×C42×C73C2311yescharacteristic
133#95×C73—21no—
134#95×C73—21no—
135#95×C73D411yescharacteristic
136#95×C73D411yescharacteristic
137C23×D73—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146C22×C412yes—
144C2×D292—21no—
145C2×D292—21no—
146C2×D292—21no—
147C2×D292—21no—
148C2×D292—21no—
149C2×D292C22×C412yes—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C2×C4×D73—42no—
154C2×C4×D73—42no—
155C2×C4×D73—81no—
156#11924018—28no—
157#11924015—28no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831C22×C412yes—
168C2×#10596831C22×C412yes—
169C2×#10596831C2×D412yes—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73C2×D412yes—
174#11924007—42no—
175#11924005—24no—
176#11924005—24no—
177#11924005—42no—
178#11924005—42no—
179#11924005—81no—
180C4×Dic73—42no—
181C4×Dic73—81no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73C22×C412yes—
186C2×Q8×C73C2×D412yes—
187C2×Q8×C73#4112yes—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199C4⋊C4×C73—24no—
200C4⋊C4×C73—22no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—41no—
207C4⋊C4×C73—42no—
208C4⋊C4×C73—42no—
209C4⋊C4×C73—42no—
210C4⋊C4×C73—81no—
211C4⋊C4×C73—81no—
212C4⋊C4×C73—81no—
213C4⋊C4×C73C22×C412yes—
214C42×C73—24no—
215C42×C73—21no—
216C42×C73—21no—
217C42×C73—21no—
218C42×C73—21no—
219C42×C73—42no—
220C42×C73—42no—
221C42×C73C22×C412yes—
222C42×C73#4112yes—
223C23×C73C22×D411yescharacteristic
224C22×D73—22no—
225C22×D73—41no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230C22×D73—41no—
231Q8×C73—22no—
232Q8×C73—22no—
233Q8×C73—22no—
234Q8×C73C4×D414yes—
235C2×C4×C73—22no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73C4×D412yes—
263C2×C4×C73C4×D412yes—
264C2×C4×C73#13712yes—
265C2×C4×C73C23×C411yescharacteristic
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×C4×C732−1+411yescharacteristic
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280D292—22no—
281D292—22no—
282D292—41no—
283D292—41no—
284C4×D73—84no—
285Dic146—22no—
286Dic146—22no—
287Dic146—41no—
288Dic146—41no—
289#10596831—22no—
290#10596831—22no—
291#10596831—22no—
292#10596831—41no—
293C22×C73C2×C4×D411yescharacteristic
294C22×C73D4211yescharacteristic
295C22×C73#54811yescharacteristic
296C22×C73C2×#13711yescharacteristic
297C22×C73#57511yescharacteristic
298C22×C73#57711yescharacteristic
299C22×C73#57711yescharacteristic
300C2×D73—42no—
301C2×D73—42no—
302C2×D73—42no—
303C2×D73—81no—
304C2×D73—81no—
305C2×D73—81no—
306C2×D73—81no—
307C4×C73—24no—
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—81no—
316C4×C73—81no—
317C4×C73C2×C4×D411yescharacteristic
318C4×C73#52011yescharacteristic
319C4×C73#54711yescharacteristic
320C4×C73#55311yescharacteristic
321Dic73—42no—
322Dic73—42no—
323Dic73—81no—
324#23013—731nomaximal, Sylow
325C2×C73#236111yescharacteristic
326C2×C73C4×#13711yescharacteristic
327C2×C73#295211yescharacteristic
328C2×C73#326611yescharacteristic
329C2×C73#327111yescharacteristic
330C2×C73#336911yescharacteristic
331C2×C73#337111yescharacteristic
332D73—82no—
333C2×#464—731no—
334C2×#441—731no—
335C2×C4×SD16—731no—
336#2367—731no—
337Q8×C4⋊C4D7311yescharacteristic, complemented
338#1959—731no—
339#1948—731no—
340#1936—731no—
341#1933—731no—
342#1880—731no—
343#1878—731no—
344#1865—731no—
345#1864—731no—
346#1771—731no—
347#1763—731no—
348C73#2301311yescharacteristic, complemented, Sylow
349C22×Q16—731no—
350C22×SD16—731no—
351C2×#124—731no—
352C2×C4×Q8—731no—
353C2×C4×Q8C2×D7311yescharacteristic
354C2×C4×Q8C2×D7311yescharacteristic
355C2×C4×Q8C2×D7311yescharacteristic
356C2×C4×D4—731no—
357#464—738no—
358#441—1464no—
359C4×SD16—1464no—
360C2×#106—731no—
361C2×#106—731no—
362C2×C4⋊C8—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#102—731no—
369C2×#102—731no—
370C2×#102—731no—
371C2×#97—731no—
372C2×#97—731no—
373C2×C4×C8—731no—
374#391—731no—
375#390—731no—
376#389—21no—
377#389—21no—
378#389C2×D7311yescharacteristic
379#389C2×D7311yescharacteristic
380#388—1461no—
381#388—1461no—
382#385—1461no—
383#384—21no—
384#384C2×D7311yescharacteristic
385C4×C4⋊C4—21no—
386C4×C4⋊C4—731no—
387C4×C4⋊C4C2×D7311yescharacteristic
388C4×C22⋊C4—1461no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#336—731no—
394#336—731no—
395C22×Q8—731no—
396C22×Q8C22×D7311yescharacteristic
397C22×D4—731no—
398C23×C4—1461no—
399C2×Q16—734no—
400C2×Q16—1462no—
401C2×Q16—1462no—
402C2×SD16—734no—
403C2×SD16—1462no—
404C2×SD16—1462no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409#124—734no—
410#124—1462no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—41no—
415C4×Q8—41no—
416C4×Q8—2921no—
417C4×Q8—1462no—
418C4×Q8C4×D7314yescomplemented
419C4×D4—2921no—
420C4×D4—1462no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4C22⋊D7311yescharacteristic
432C2×C4⋊C4C22⋊D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42C22×D7311yescharacteristic
447C2×C42C22×D7311yescharacteristic
448C2×C42C22×D7311yescharacteristic
449#106—1462no—
450#106—1462no—
451C4⋊C8—734no—
452#103—2921no—
453#103—734no—
454#103—734no—
455#103—1462no—
456#103—1462no—
457#102—2921no—
458#102—734no—
459#102—1462no—
460#97—1462no—
461#97—1462no—
462C4×C8—1462no—
463#95—21no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95C22⋊D7311yescharacteristic
469#95C22⋊D7311yescharacteristic
470C24—1461no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—732no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8C2×C4×D7312yes—
481C2×Q8D4×D7312yes—
482C2×Q8#1192402912yes—
483C2×D4—732no—
484C2×D4—1461no—
485C2×D4—1461no—
486C2×D4—1461no—
487C2×D4—1461no—
488C2×D4—1461no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—2922no—
494C22×C4—2922no—
495C22×C4—5841no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C2×C22⋊D7311yescharacteristic
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C23×D7311yescharacteristic
506Q16—1468no—
507SD16—1468no—
508C2×C8—732no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—2922no—
535C4⋊C4—732no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4C2×C4×D7312yes—
540C22⋊C4—2922no—
541C22⋊C4—2922no—
542C22⋊C4—5841no—
543C22⋊C4—1464no—
544C22⋊C4—1464no—
545C42—24no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—42no—
552C42—2922no—
553C42—5841no—
554C42C2×C4×D7312yes—
555C42#1192402812yes—
556C23—2921no—
557C23—2921no—
558C23—2921no—
559C23—2921no—
560C23—2921no—
561C23—2921no—
562C23—1462no—
563C23C22×C22⋊D7311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—2921no—
568Q8—2921no—
569Q8—1462no—
570Q8—1462no—
571Q8#1455655314yes—
572D4—2921no—
573D4—2921no—
574D4—1462no—
575D4—1462no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—5844no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4C4×C22⋊D7312yes—
614C2×C4C4×C22⋊D7312yes—
615C2×C4#1455662212yes—
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C2×D4×D7311yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C2×C4C2×#1192402911yescharacteristic
620C2×C4#1455663611yescharacteristic
621C2×C4#1455663911yescharacteristic
622C8—2921no—
623C8—1462no—
624C8—1462no—
625C8—1462no—
626C22—2922no—
627C22—2922no—
628C22—2922no—
629C22—5841no—
630C22—5841no—
631C22—5841no—
632C22—5841no—
633C22C2×C4×C22⋊D7311yescharacteristic
634C22D4×C22⋊D7311yescharacteristic
635C22#1798924811yescharacteristic
636C22C2×#1455662211yescharacteristic
637C22#1798938511yescharacteristic
638C22#1798939511yescharacteristic
639C22#1798939811yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—2922no—
651C4—2922no—
652C4—5841no—
653C4C2×#1455655311yescharacteristic
654C4#1798906811yescharacteristic
655C4#1798923811yescharacteristic
656C4#1798928511yescharacteristic
657C2—5842no—
658C2#2261294711yescharacteristic
659C2C4×#1455662211yescharacteristic
660C2#2261881211yescharacteristic
661C2#2262144211yescharacteristic
662C2#2262151211yescharacteristic
663C2#2262267111yescharacteristic
664C2#2262269911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingQ8×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterQ8×C4⋊C4D73
hypocenterC73#23013
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17988692C2semidirect
6. #22610692C2semidirect
8. #22610606C2semidirect
10. #22610272C2semidirect
11. #22610198C2semidirect
13. #22609806C2semidirect
14. #22609761C2semidirect
15. Q8×C4⋊C4×C73C2semidirect
21. #17988809C4semidirect
67. C2×#11924018C2×C4semidirect
93. C4⋊#10596831C2×C4semidirect
98. #14556514C2×C4semidirect
114. C4×Q8×C73C2×C4semidirect
337. Q8×C4⋊C4D73semidirect
348. C73#23013semidirect
418. C4×Q8C4×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888092
C2×#179886922
C2×C4×#119240152
#226130002
#226107452
#226106922
#226106272
#226106062
#226102862
#226102722
#226101982
#226101912
#226098062
#226097612
Q8×C4⋊C4×C732
#2301373